Applying free random variables to random matrix analysis of financial data. Part I: The Gaussian case

We apply the concept of free random variables to doubly correlated (Gaussian) Wishart random matrix models, appearing, for example, in a multivariate analysis of financial time series, and displaying both inter-asset cross-covariances and temporal auto-covariances. We give a comprehensive introduction to the rich financial reality behind such models. We explain in an elementary way the main techniques of free random variables calculus, with a view to promoting them in the quantitative finance community. We apply our findings to tackle several financially relevant problems, such as a universe of assets displaying exponentially decaying temporal covariances, or the exponentially weighted moving average, both with an arbitrary structure of cross-covariances.

[1]  Gilles O. Zumbach,et al.  Volatility processes and volatility forecast with long memory , 2002 .

[2]  Mantegna,et al.  Taxonomy of stock market indices , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  S. Simon,et al.  Eigenvalue density of correlated complex random Wishart matrices. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  E. J. Gumbel,et al.  Statistics of Extremes. , 1960 .

[5]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[6]  Signal and noise in financial correlation matrices , 2003, cond-mat/0312496.

[7]  Free Extreme Values , 2005, math/0501274.

[8]  T. W. Epps.  Comovements in Stock Prices in the Very Short Run , 1979 .

[9]  Philippe Biane,et al.  Free diffusions, free entropy and free Fisher information , 2001 .

[10]  Yannick Malevergne,et al.  Collective origin of the coexistence of apparent random matrix theory noise and of factors in large sample correlation matrices , 2002, cond-mat/0210115.

[11]  Imre Kondor,et al.  Estimated correlation matrices and portfolio optimization , 2003, cond-mat/0305475.

[12]  Noh.  Model for correlations in stock markets , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  Analysis of delay correlation matrices , 2006, cond-mat/0601279.

[14]  J. Bouchaud,et al.  Theory Of Financial Risk And Derivative Pricing , 2000 .

[15]  M. Potters,et al.  Exponential Weighting and Random-Matrix-Theory-Based Filtering of Financial Covariance Matrices for Portfolio Optimization , 2004, cond-mat/0402573.

[16]  V. Plerou,et al.  Universal and Nonuniversal Properties of Cross Correlations in Financial Time Series , 1999, cond-mat/9902283.

[17]  M. Fréchet.  Sur la loi de probabilité de l'écart maximum , 1928 .

[18]  Stefan Thurner,et al.  Random matrix ensembles of time-lagged correlation matrices: derivation of eigenvalue spectra and analysis of financial time-series , 2006 .

[19]  Yan V. Fyodorov.  Recent Perspectives in Random Matrix Theory and Number Theory: Introduction to the random matrix theory: Gaussian Unitary Ensemble and beyond , 2005 .

[20]  Ronald L. Graham,et al.  Concrete mathematics - a foundation for computer science , 1991 .

[21]  David Tse,et al.  Linear Multiuser Receivers: Effective Interference, Effective Bandwidth and User Capacity , 1999, IEEE Trans. Inf. Theory.

[22]  Z. Burda,et al.  Spectral properties of empirical covariance matrices for data with power-law tails. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Z. Burda,et al.  Levy Matrices and Financial Covariances , 2001 .

[24]  K. Shadan,et al.  Available online: , 2012 .

[25]  E. Gumbel,et al.  Les valeurs extrêmes des distributions statistiques , 1935 .

[26]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[27]  Jean-Philippe Bouchaud,et al.  Financial Applications of Random Matrix Theory: Old Laces and New Pieces , 2005 .

[28]  J. Bouchaud,et al.  RANDOM MATRIX THEORY AND FINANCIAL CORRELATIONS , 2000 .

[29]  C. Tracy,et al.  Introduction to Random Matrices , 1992, hep-th/9210073.

[30]  Ralf R. Müller,et al.  A random matrix model of communication via antenna arrays , 2002, IEEE Trans. Inf. Theory.

[31]  G. Hooft.  A Planar Diagram Theory for Strong Interactions , 1974 .

[32]  B. Gnedenko.  Sur La Distribution Limite Du Terme Maximum D'Une Serie Aleatoire , 1943 .

[33]  W. Sharpe.  CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK* , 1964 .

[34]  A. Zee.  Law of addition in random matrix theory , 1996 .

[35]  Jaroslaw Kwapien,et al.  Quantifying the dynamics of financial correlations , 2001 .

[36]  Heidelberg,et al.  A New Method to Estimate the Noise in Financial Correlation Matrices , 2002, cond-mat/0206577.

[37]  2D QUANTUM GRAVITY,MATRIX MODELS AND GRAPH COMBINATORICS , 2004, math-ph/0406013.

[38]  Anne Boutet de Monvel,et al.  Some Elementary Results around the Wigner Semicircle Law , 2001 .

[39]  The planar sector of field theories , 1982 .

[40]  Jens Svensson.  The asymptotic spectrum of the EWMA covariance estimator , 2007 .

[41]  R. Fisher,et al.  Limiting forms of the frequency distribution of the largest or smallest member of a sample , 1928, Mathematical Proceedings of the Cambridge Philosophical Society.

[42]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[43]  Jorge Mina,et al.  Return to RiskMetrics: The Evolution of a Standard , 2001 .

[44]  Maciej A. Nowak,et al.  Non-Hermitian random matrix models: Free random variable approach , 1997 .

[45]  J. Wishart.  THE GENERALISED PRODUCT MOMENT DISTRIBUTION IN SAMPLES FROM A NORMAL MULTIVARIATE POPULATION , 1928 .

[46]  D. Voiculescu.  Limit laws for Random matrices and free products , 1991 .

[47]  Imre Kondor,et al.  Noisy covariance matrices and portfolio optimization , 2002 .

[48]  J. Bouchaud,et al.  Rational Decisions, Random Matrices and Spin Glasses , 1998, cond-mat/9801209.

[49]  J. W. Silverstein,et al.  On the empirical distribution of eigenvalues of a class of large dimensional random matrices , 1995 .

[50]  P. A. P. Moran,et al.  An introduction to probability theory , 1968 .

[51]  János Kertész,et al.  Modeling the Epps effect of cross correlations in asset prices , 2007, SPIE International Symposium on Fluctuations and Noise.

[52]  Z. Burda,et al.  Spectral moments of correlated Wishart matrices. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  V. Marčenko,et al.  DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES , 1967 .

[54]  J. Hull.  Options, Futures, and Other Derivatives , 1989 .

[55]  Z. Burda,et al.  Is Econophysics a Solid Science , 2003, cond-mat/0301096.

[56]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[57]  Antonia Maria Tulino,et al.  Random Matrix Theory and Wireless Communications , 2004, Found. Trends Commun. Inf. Theory.

[58]  Z. Burda,et al.  Signal and Noise in Correlation Matrix , 2003, cond-mat/0305627.

[59]  Jerzy Jurkiewicz,et al.  Free random Lévy and Wigner-Lévy matrices. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[60]  Various shades of blue's functions , 1997, hep-th/9710103.

[61]  J. H. Lint.  Concrete mathematics : a foundation for computer science / R.L. Graham, D.E. Knuth, O. Patashnik , 1990 .

[62]  P. J. Forrester,et al.  Developments in random matrix theory , 2003, cond-mat/0303207.

[63]  H. Iemoto.  Modelling the persistence of conditional variances , 1986 .

[64]  J. Jurkiewicz,et al.  On diffusion of large matrices , 2005 .

[65]  R. Speicher.  Multiplicative functions on the lattice of non-crossing partitions and free convolution , 1994 .

[66]  Ralf R. Muller.  A random matrix model of communication via antenna arrays , 2002 .

[67]  V. Plerou,et al.  Random matrix approach to cross correlations in financial data. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[68]  Eigenvalue density of empirical covariance matrix for correlated samples , 2005, cond-mat/0508451.

[69]  J. Bouchaud,et al.  Large dimension forecasting models and random singular value spectra , 2005, physics/0512090.

[70]  T. Guhr,et al.  RANDOM-MATRIX THEORIES IN QUANTUM PHYSICS : COMMON CONCEPTS , 1997, cond-mat/9707301.

[71]  W. Weibull.  A Statistical Distribution Function of Wide Applicability , 1951 .

[72]  G. Biroli,et al.  The Student ensemble of correlation matrices: eigenvalue spectrum and Kullback-Leibler entropy , 2007, 0710.0802.

[73]  Gilles O. Zumbach,et al.  The Riskmetrics 2006 Methodology , 2007 .

[74]  Gregory Connor,et al.  How much structure is best? A comparison of market model, factor model and unstructured equity covariance matrices , 2008 .

[75]  The Wick theorem for non-Gaussian distributions and its application for noise filtering of correlated q-Exponentialy distributed random variables , 2004, math-ph/0411020.

[76]  MASTERING THE MASTER FIELD , 1994, hep-th/9411021.

[77]  S. Péché,et al.  Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices , 2004, math/0403022.

[78]  Z. Burda,et al.  Free random Lévy matrices. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[79]  Z. Burda,et al.  Free Lévy matrices and financial correlations , 2001, cond-mat/0103109.

[80]  STABLE LAWS AND DOMAINS OF ATTRACTION IN FREE PROBABILITY THEORY , 1999, math/9905206.

[81]  R. Speicher,et al.  Spectra of Hamiltonians with generalized single-site dynamical disorder , 1994, cond-mat/9404037.

[82]  O. Johnson.  Free Random Variables , 2004 .

[83]  Maciej A. Nowak,et al.  Random matrix filtering in portfolio optimization , 2005 .

[84]  E. Wigner.  Characteristic Vectors of Bordered Matrices with Infinite Dimensions I , 1955 .

[85]  J. Bouchaud,et al.  Noise Dressing of Financial Correlation Matrices , 1998, cond-mat/9810255.

[86]  Free random Lévy variables and financial probabilities , 2001, cond-mat/0103140.

[87]  Romuald A. Janik,et al.  Multiplying unitary random matrices—universality and spectral properties , 2003 .

[89]  M. Oshikawa,et al.  Random matrix theory analysis of cross correlations in financial markets. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[90]  I. Kondor,et al.  Noisy Covariance Matrices and Portfolio Optimization II , 2002, cond-mat/0205119.

[91]  M. Tumminello,et al.  Hierarchically nested factor model from multivariate data , 2007 .